math.log/2 on complex, real part

Percentage Accurate: 51.6% → 99.3%

Time: 25.4s

Alternatives: 4

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (/
  (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0))
  (+ (* (log base) (log base)) (* 0.0 0.0))))

C

double code(double re, double im, double base) {
	return ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
}

Fortran

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0d0)) / ((log(base) * log(base)) + (0.0d0 * 0.0d0))
end function

Java

public static double code(double re, double im, double base) {
	return ((Math.log(Math.sqrt(((re * re) + (im * im)))) * Math.log(base)) + (Math.atan2(im, re) * 0.0)) / ((Math.log(base) * Math.log(base)) + (0.0 * 0.0));
}

Python

def code(re, im, base):
	return ((math.log(math.sqrt(((re * re) + (im * im)))) * math.log(base)) + (math.atan2(im, re) * 0.0)) / ((math.log(base) * math.log(base)) + (0.0 * 0.0))

Julia

function code(re, im, base)
	return Float64(Float64(Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) * log(base)) + Float64(atan(im, re) * 0.0)) / Float64(Float64(log(base) * log(base)) + Float64(0.0 * 0.0)))
end

MATLAB

function tmp = code(re, im, base)
	tmp = ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
end

Wolfram

code[re_, im_, base_] := N[(N[(N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(N[ArcTan[im / re], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Log[base], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(0.0 * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 51.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (/
  (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0))
  (+ (* (log base) (log base)) (* 0.0 0.0))))

C

double code(double re, double im, double base) {
	return ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
}

Fortran

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0d0)) / ((log(base) * log(base)) + (0.0d0 * 0.0d0))
end function

Java

public static double code(double re, double im, double base) {
	return ((Math.log(Math.sqrt(((re * re) + (im * im)))) * Math.log(base)) + (Math.atan2(im, re) * 0.0)) / ((Math.log(base) * Math.log(base)) + (0.0 * 0.0));
}

Python

def code(re, im, base):
	return ((math.log(math.sqrt(((re * re) + (im * im)))) * math.log(base)) + (math.atan2(im, re) * 0.0)) / ((math.log(base) * math.log(base)) + (0.0 * 0.0))

Julia

function code(re, im, base)
	return Float64(Float64(Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) * log(base)) + Float64(atan(im, re) * 0.0)) / Float64(Float64(log(base) * log(base)) + Float64(0.0 * 0.0)))
end

MATLAB

function tmp = code(re, im, base)
	tmp = ((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
end

Wolfram

code[re_, im_, base_] := N[(N[(N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(N[ArcTan[im / re], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Log[base], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] + N[(0.0 * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \log base}{\log base \cdot \log base} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (/ (* (log (hypot re im)) (log base)) (* (log base) (log base))))

C

double code(double re, double im, double base) {
	return (log(hypot(re, im)) * log(base)) / (log(base) * log(base));
}

Java

public static double code(double re, double im, double base) {
	return (Math.log(Math.hypot(re, im)) * Math.log(base)) / (Math.log(base) * Math.log(base));
}

Python

def code(re, im, base):
	return (math.log(math.hypot(re, im)) * math.log(base)) / (math.log(base) * math.log(base))

Julia

function code(re, im, base)
	return Float64(Float64(log(hypot(re, im)) * log(base)) / Float64(log(base) * log(base)))
end

MATLAB

function tmp = code(re, im, base)
	tmp = (log(hypot(re, im)) * log(base)) / (log(base) * log(base));
end

Wolfram

code[re_, im_, base_] := N[(N[(N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] / N[(N[Log[base], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.3%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. accelerator-lowering-hypot.f64 99.2

      \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

4. Applied egg-rr 99.2%

   \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
5. Step-by-step derivation

   1. mul0-rgt 99.2

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \log base + \color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]

6. Applied egg-rr 99.2%

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \log base + \color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]
7. Step-by-step derivation

   1. metadata-eval 99.2

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \log base + 0}{\log base \cdot \log base + \color{blue}{0}} \]

8. Applied egg-rr 99.2%

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \log base + 0}{\log base \cdot \log base + \color{blue}{0}} \]

9. Final simplification 99.2%

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \log base}{\log base \cdot \log base} \]
10. Add Preprocessing

Alternative 2: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \log base}{{\log base}^{2}} \end{array} \]

FPCore

(FPCore (re im base)
 :precision binary64
 (/ (* (log (hypot re im)) (log base)) (pow (log base) 2.0)))

C

double code(double re, double im, double base) {
	return (log(hypot(re, im)) * log(base)) / pow(log(base), 2.0);
}

Java

public static double code(double re, double im, double base) {
	return (Math.log(Math.hypot(re, im)) * Math.log(base)) / Math.pow(Math.log(base), 2.0);
}

Python

def code(re, im, base):
	return (math.log(math.hypot(re, im)) * math.log(base)) / math.pow(math.log(base), 2.0)

Julia

function code(re, im, base)
	return Float64(Float64(log(hypot(re, im)) * log(base)) / (log(base) ^ 2.0))
end

MATLAB

function tmp = code(re, im, base)
	tmp = (log(hypot(re, im)) * log(base)) / (log(base) ^ 2.0);
end

Wolfram

code[re_, im_, base_] := N[(N[(N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision] * N[Log[base], $MachinePrecision]), $MachinePrecision] / N[Power[N[Log[base], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.3%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. accelerator-lowering-hypot.f64 99.2

      \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

4. Applied egg-rr 99.2%

   \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
5. Step-by-step derivation

   1. mul0-rgt 99.2

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \log base + \color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]

6. Applied egg-rr 99.2%

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \log base + \color{blue}{0}}{\log base \cdot \log base + 0 \cdot 0} \]
7. Step-by-step derivation

   1. metadata-eval 99.2

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \log base + 0}{\log base \cdot \log base + \color{blue}{0}} \]

8. Applied egg-rr 99.2%

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \log base + 0}{\log base \cdot \log base + \color{blue}{0}} \]
9. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \log base + 0}{\color{blue}{{\log base}^{2}} + 0} \]

   2. pow-lowering-pow.f64 N/A

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \log base + 0}{\color{blue}{{\log base}^{2}} + 0} \]

   3. log-lowering-log.f64 99.2

      \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \log base + 0}{{\color{blue}{\log base}}^{2} + 0} \]

10. Applied egg-rr 99.2%

    \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \log base + 0}{\color{blue}{{\log base}^{2}} + 0} \]

11. Final simplification 99.2%

    \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \log base}{{\log base}^{2}} \]
12. Add Preprocessing

Alternative 3: 26.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \frac{\log im}{\log base} \end{array} \]

FPCore

(FPCore (re im base) :precision binary64 (/ (log im) (log base)))

C

double code(double re, double im, double base) {
	return log(im) / log(base);
}

Fortran

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = log(im) / log(base)
end function

Java

public static double code(double re, double im, double base) {
	return Math.log(im) / Math.log(base);
}

Python

def code(re, im, base):
	return math.log(im) / math.log(base)

Julia

function code(re, im, base)
	return Float64(log(im) / log(base))
end

MATLAB

function tmp = code(re, im, base)
	tmp = log(im) / log(base);
end

Wolfram

code[re_, im_, base_] := N[(N[Log[im], $MachinePrecision] / N[Log[base], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.3%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
2. Add Preprocessing

3. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\frac{\log im}{\log base}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\log im}{\log base}} \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \frac{\color{blue}{\log im}}{\log base} \]

   3. log-lowering-log.f64 24.3

      \[\leadsto \frac{\log im}{\color{blue}{\log base}} \]

5. Simplified 24.3%

   \[\leadsto \color{blue}{\frac{\log im}{\log base}} \]
6. Add Preprocessing

Alternative 4: 3.1% accurate, 562.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (re im base) :precision binary64 0.0)

C

double code(double re, double im, double base) {
	return 0.0;
}

Fortran

real(8) function code(re, im, base)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8), intent (in) :: base
    code = 0.0d0
end function

Java

public static double code(double re, double im, double base) {
	return 0.0;
}

Python

def code(re, im, base):
	return 0.0

Julia

function code(re, im, base)
	return 0.0
end

MATLAB

function tmp = code(re, im, base)
	tmp = 0.0;
end

Wolfram

code[re_, im_, base_] := 0.0

Derivation

1. Initial program 49.3%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. div-inv N/A

      \[\leadsto \color{blue}{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0\right) \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0}} \]

   2. mul0-rgt N/A

      \[\leadsto \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \color{blue}{0}\right) \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0} \]

   3. metadata-eval N/A

      \[\leadsto \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \color{blue}{\left(\mathsf{neg}\left(0\right)\right)}\right) \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0} \]

   4. sub-neg N/A

      \[\leadsto \color{blue}{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base - 0\right)} \cdot \frac{1}{\log base \cdot \log base + 0 \cdot 0} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{\log base \cdot \log base + 0 \cdot 0} \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base - 0\right)} \]

   6. metadata-eval N/A

      \[\leadsto \frac{1}{\log base \cdot \log base + \color{blue}{0}} \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base - 0\right) \]

   7. +-rgt-identity N/A

      \[\leadsto \frac{1}{\color{blue}{\log base \cdot \log base}} \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base - 0\right) \]

   8. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{\log base}}{\log base}} \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base - 0\right) \]

   9. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{\log base} \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base - 0\right)}{\log base}} \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{\log base} \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base - 0\right)}{\log base}} \]

4. Applied egg-rr 49.3%

   \[\leadsto \color{blue}{\frac{\frac{1}{\log base} \cdot \mathsf{fma}\left(0.5 \cdot \log \left(\mathsf{fma}\left(re, re, im \cdot im\right)\right), \log base, 0\right)}{\log base}} \]

5. Taylor expanded in re around 0

   \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left({im}^{2}\right)}}{\log base} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left({im}^{2}\right)}}{\log base} \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\log \left({im}^{2}\right)}}{\log base} \]

   3. unpow2 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \log \color{blue}{\left(im \cdot im\right)}}{\log base} \]

   4. *-lowering-*.f64 24.8

      \[\leadsto \frac{0.5 \cdot \log \color{blue}{\left(im \cdot im\right)}}{\log base} \]

7. Simplified 24.8%

   \[\leadsto \frac{\color{blue}{0.5 \cdot \log \left(im \cdot im\right)}}{\log base} \]
8. Step-by-step derivation

   1. log-prod N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left(\log im + \log im\right)}}{\log base} \]

   2. distribute-rgt-in N/A

      \[\leadsto \frac{\color{blue}{\log im \cdot \frac{1}{2} + \log im \cdot \frac{1}{2}}}{\log base} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\color{blue}{\log im \cdot \left(\frac{1}{2} + \frac{1}{2}\right)}}{\log base} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\log im \cdot \color{blue}{1}}{\log base} \]

   5. *-rgt-identity N/A

      \[\leadsto \frac{\color{blue}{\log im}}{\log base} \]

   6. *-lft-identity N/A

      \[\leadsto \color{blue}{1 \cdot \frac{\log im}{\log base}} \]

   7. *-inverses N/A

      \[\leadsto \color{blue}{\frac{\log base}{\log base}} \cdot \frac{\log im}{\log base} \]

   8. times-frac N/A

      \[\leadsto \color{blue}{\frac{\log base \cdot \log im}{\log base \cdot \log base}} \]

   9. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\log base \cdot \log im\right)}{\mathsf{neg}\left(\log base \cdot \log base\right)}} \]

9. Applied egg-rr 3.1%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024194 
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  :precision binary64
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))